Categories, groupoids, pseudogroups and analytical structures

CONTENTS Introduction................................................................................................................................................. 3 I. TERMS AND NOTATION....................................................................................................................... 5 II. GROUPOIDS AND CATEGORIES...................................................................................................... 6 1. The notion of groupoid......................................................................................................................... 6 2. Equivalence of the definition of groupoid to the definition of Ehresmann.................................. 8 3. Relationship between l.lio notion of an liliroHiminn groupoid and the notion of a Brandt, groupoid...................................................................................................................................................... 9 4. Categories of functions and representation theorems................................................................. 12 5. The algebraic product of sets and the closure of a sot in the multiplicative system............... 15 III. THE RELATIONSHIP BETWEEN A GOŁĄB PSEUDOOROUP AND AN EHRESMANN GROUPOID............................................................................................................... 16 6. The notions of a Gołąb pseudogroup and of a functional element............................................ 16 7. The isomorphism of an arbitrary Ehresmann groupoid and a Gołąb pseudogroup of a certain type. Groupoids of functional elements................................................. 18 IV. GENERATING IN GOŁĄB PSEUDOGROUPS AND SOME PROPERTIES OF A SET OF FUNCTIONS.............................................................................................................................. 20 8. Some operations with sets of functions........................................................................................... 21 9. A quasi-order of the family of all subsets of the set, L (X)............................................................. 24 10. Determining a pseudogroups with the aid of sets of functional elements............................. 26 11. The problom of the existence of the smallest pseudogroup including a given set of local homeomorphisms...................................................................................................................... 29 V. SEMI-PSEUDOGROUPS AND A GENERALIZATION OP THE NOTION OF AN ANALYTICAL STRUCTURE................................................................................................................ 33 12. Semi-pseudogroups......................................................................................................................... 33 13. The notion of an analytical structure............................................................................................... 35 References................................................................................................................................................. 39